Incidence bounds in positive characteristic via valuations and distality

Abstract

We prove distality of quantifier-free relations on valued fields with finite residue field. By a result of Chernikov-Galvin-Starchenko, this yields Szemerédi-Trotter-like incidence bounds for function fields over finite fields. We deduce a version of the Elekes-Szabó theorem for such fields.

v2: local fixes; v3: local clarifications (published version); v4: fix minor error with parameters in defn of "distal in"